Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 202 / 4 / d

As printed, the requested conclusion is false for $\delta \ne 1$. On an interval on which $X$ stays positive, the time change of a continuous process satisfies $d{\tau }_s=ds/({\delta }^2{X}_s^{2(\delta -1)/\delta })$. The time-changed martingale term has quadratic variation $s$, so the Lévy characterization of Brownian motion identifies it with a standard Brownian motion $\tilde{B}$. Dividing the drift in part (a) by the derivative of the clock givesConsequently the construction actually satisfieswhich is the Bessel process equation of dimension $2-1/\delta$. It equals the paper's claimed drift $(\delta -1)/(2X_s)$ only when $\delta =1$. The mismatch between the specified power, clock, and conclusion is therefore a typographical error in the question.